Convergence of Discrete Dirichlet Forms to Continuous Dirichlet Forms on Fractals |
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Authors: | Peirone Roberto |
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Institution: | (1) Dipartimento di Matematica, Università di Roma Tor Vergata , via della Ricerca Scientifica, 00133 Roma, Italy (e-mail |
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Abstract: | It is well known that a Dirichlet form on a fractal structure can be defined as the limit of an increasing sequence of discrete Dirichlet forms, defined on finite subsets which fill the fractal. The initial form is defined on V
(0), which is a sort of boundary of the fractal, and we have to require that it is an eigenform, i.e., an eigenvector of a particular nonlinear renormalization map for Dirichlet forms on V
(0). In this paper, I prove that, provided an eigenform exists, even if the form on V
(0) is not an eigenform, the corresponding sequence of discrete forms converges to a Dirichlet form on all of the fractal, both pointwise and in the sense of -convergence (but these two limits can be different). The problem of -convergence was first studied by S. Kozlov on the Gasket. |
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Keywords: | fractals Dirichlet forms and pointwise convergence" target="_blank">gif" alt="Gamma" align="BASELINE" BORDER="0"> and pointwise convergence |
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